Some yeasts reproduce through fission, in which one cell splits into two daughter cells, while other yeasts use budding, a process involving the formation of a small growth on a parental cell. The nucleus of that cell divides, and the new nucleus moves into the bud, which grows and eventually becomes a separate cell. Other yeasts use sexual reproduction through the joining of cells and fusing nuclei to create a new diploid cell, which starts budding and making its own colony of cells.
Fission and budding are the asexual forms of yeast reproduction, and only diploid cells reproduce in this way. In general terms, this process is quite similar to general mitosis. The new cells are precise copies of the parent cells and can either be haploid or diploid.
Only haploid yeast cells reproduce sexually, and when they mate, generally they are not of the same gender. Before reproduction, both haploids go through shmooing, in which the cells lengthen and become thinner. When they meet, they fuse and join nuclei in order to create a new diploid. When environmental conditions turn hostile, diploids split into haploid spores, making future sexual reproduction possible.
For yeast to reproduce, the environment must be at least somewhat moist and lukewarm, and the cells need access to sugar. When bread rises, the cause is reproduction of yeast cells within the dough.